For any topic related to matrices. This includes: systems of linear equations, eigenvalues and eigenvectors (diagonalization, triangularization), determinant, trace, characteristic polynomial, adjugate, transpose, Jordan normal form, matrix algorithms (e.g. LU, Gauss elimination, SVD, QR), ...

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2
votes
2answers
29 views

If we have the transformation of 2 vectors, how can we define the transformation?

I know how to find the transformation matrix if I have the definition of the transformation with respect to the basis vectors. Now, I am given 2 vectors and its transformations, and I need to define ...
2
votes
1answer
166 views

determinant of infinitely large matrix by decomposition

Read the too long didnt read version in bold before going into the finer detail. The overall point is that when I decompose this matrix to try and find its determinant I get an answer that doesn't ...
5
votes
1answer
68 views

Linear Transformation Matrix with polynomials

A linear transformation $T : P_2 \to P_2$ has matrix with respect to $S$ given by: $$[T]\,( S) = \begin{bmatrix} 1/2&-3&1/2\\ -1&4&-1\\ 1/2&2&1/2\\ \end{bmatrix} $$ How do ...
0
votes
1answer
53 views

Show $X^{-1}AX$ is an upper Hessenberg matrix

Let $A \in \mathbb{C}^{n\times n}$, $x \in \mathbb{C}^n$, $X=[x,Ax,A^2x,\ldots,A^{n-1}x]$ and let $X$ be non-singular. Show that $X^{-1}AX$ is an upper Hessenberg matrix.
1
vote
0answers
34 views

Matrix $A$ has random integer entries. Probability to get a matrix of integer elements after $A$ has been echelon row reduced?

Let $A$ be an $n \times m$ matrix with integer elements uniformly randomly chosen between $-k$ and $k$. What is the probability to get a matrix of integer elements after $A$ has been reduced in ...
1
vote
0answers
46 views

Why row operations do not change the columns space?

I understand why the elementary row operations do not change the row space, but why they do not change the column space (or at least the dependence between the columns)?
0
votes
3answers
46 views

Prove that $A$ is regular matrix and find $A^{-1}$ if $A,I\in M_{n\times n}(\mathbb{R})$ and $(A+I)^3=O$

Expansion of a binomial gives: $$A^3+3A^2+3A+I=0$$ We know that the matrix is regular if squared and has inverse ($detA\neq 0$). Is it possible to determine from above equation that $detA\neq 0$?
4
votes
1answer
182 views

Matrices $B$ that commute with every matrix commuting with $A$

There have been many questions in the vein of this one, but I can't find one that answers it specifically. Suppose $A,B\in M_n(\mathbb C)$ are two matrices such that, for any other matrix $C\in ...
1
vote
2answers
54 views

Prove that $A^{-1}=A^n$ if $A^n+A^{n-1}+…+A+E=O,n\in \mathbb{N}$ and $A$ is regular matrix

I took the example of $2\times 2$ matrix $$ \begin{bmatrix} x & y \\ u & v \\ \end{bmatrix}$$ which gives matrix equation: $$A^2+A+E=O$$ After addition, I get system ...
6
votes
2answers
76 views

Order $n^2$ different reals, such that they form a $\mathbb{R^n}$ basis

I've been trying to solve this linear algebra problem: You are given $n^2 > 1$ pairwise different real numbers. Show that it's always possible to construct with them a basis for $\mathbb{R^n}$. ...
0
votes
1answer
1k views

find a change of basis matrix

Given the basis $\beta = \{(1, 1, 0),\ (1, 0, -1),\ (2, 1, 0)\}$ and matrix $$ A = \begin{bmatrix} 1 & 1 & 0 \\ 1 & 1 & 0 \\ 0 & 0 & 1 \\ ...
1
vote
1answer
145 views

Is there any advantage using matrices instead of equations do describe a trochoid?

I was reading this post at Physics SE, where a lengthy explanation is given to describe a trochoid, the combination of linear and rotational motion of a rod. Can you explain what is the point/ ...
3
votes
2answers
183 views

Show determinant of $\left[\begin{matrix} A & 0 \\ C & D\end{matrix}\right] = \det{A}\cdot \det{D}$

Let $A \in \mathbb{R}^{n, n}$, $B \in \mathbb{R}^{n, m}$, $C \in \mathbb{R}^{m, n}$ and $D \in \mathbb{R}^{m, m}$ be matrices. Now, I have seen on Wikipedia the explanation of why determinant of ...
0
votes
1answer
164 views

The nullity of a square matrix with linearly dependent rows is at least one.

The nullity of a square matrix with linearly dependent rows is at least one. True or False? Here is the answer my textbook gives: True; if the rows are linearly dependent, then the rank is at ...
1
vote
1answer
36 views

Matrix Decomposition Question: If $A$ is symmetric when is it true that $A = B^T B $

I was thinking about the fact that for a symmetric matrix A the maximum values of $X^T A X$, where $\|X\| = 1$ occurs when $X$ is an eigenvalue of $A$. I was trying to think of a good geometric image ...
0
votes
2answers
29 views

Computing determinants

While computing a determinant I know I can do the following: a) factor out common constants (and in doings so multiply the determinant by the constant) b) add rows (or columns) or a constant times a ...
0
votes
0answers
17 views

A better way to visualize the change of output with respect to input of 3-dimentional data sets?

Background (1) I have a system $Ax=b$. Let us assume we do not know any information about $A$. (2) Both $x$ and $b$ are 3 dimensional data. In terms of physical meanings, both of them describes ...
0
votes
1answer
562 views

Inverse of a rigid transformation

I would be grateful for any help with the steps required to complete this calculation. You may assume that I have some experience with matrices from before, but I am obviously no master! So we have ...
5
votes
3answers
376 views

Visual representation of matrices

I am used to seeing most basic mathematical objects being visually represented (for instance, a curve in the plane divided by the xy axis; the same goes for complex numbers, vectors, and so on....), ...
1
vote
1answer
20 views

Matrix representation in exponential form

So having worked out beforehand that $Λ(v) = \begin{pmatrix} γ&0&\frac{-γv}{c}\\ 0&1&0\\\frac{-γv}{c}&0&γ\end{pmatrix}$ where $Λ(v) ∈ SO(2,1)$ is a matrix representation of a ...
2
votes
1answer
99 views

How to turn this matrix to Jordan normal form?

Matrix $A$ is $ \left( \begin{array}{ccc} 3 & 0 & 8 \\ 3 & -1 & 6 \\ -2 & 0 & -5 \end{array} \right)$ and I need to find a matrix P such that $P^{-1} A P = J$ where $J$ is a ...
2
votes
1answer
207 views

Explicit Calculation of Banded Toeplitz Matrix Eigenvalues

I recently found a paper which detailed a method of finding the eigenvalues of the $n\times n$ banded Toeplitz matrix $$ \left[ \begin{array}{ccccccc} a_0 & a_1 & a_2 & \dots & a_s ...
3
votes
4answers
87 views

Show that $A$ and $A^T$ do not have the same eigenvectors in general

I understood that $A$ and $A^T$ have the same eigenvalues, since $$\det(A - \lambda I)= \det(A^T - \lambda I) = \det(A - \lambda I)^T$$ The problem is to show that $A$ and $A^T$ do not have the same ...
1
vote
1answer
46 views

If $\vec{v}$ is an eigenvector of $A$, then also $B\vec{v}$, when $AB = BA$ [duplicate]

I have the following problem: Let $V$ be a finite dimensional vector space. Let $A$, $B$ be linear maps of $V$ into itself. Assume that $AB = BA$. Show that if $\vec{v}$ is an eigenvector of $A$, ...
0
votes
1answer
34 views

Formula for transpose matrix multiplication

I ran by an exercise in my textbook that got me writing sheets of paper of matrix multiplications, and I got a lot of ($ij$ entry)$^2$, but that was not enough for me to construct a convincing ...
2
votes
4answers
10k views

Help demystify the Navy PFA equations.

I need help finding an equation that the Navy's Physical Readiness Program Office (PRIMS) keeps unpublished for some unexplainable reason and will not share after numorous requests. Anyways, luckily, ...
1
vote
1answer
53 views

What is the spectrum of this matrix?

$$A_n=\begin{bmatrix} 1 & 1 & 1 & \cdots & 1 & 1\\ 1 & 2 & 2 & \cdots & 2 & 2\\ 1 & 2 & 3 & \cdots & 3 & 3\\ \vdots & \vdots & ...
1
vote
3answers
109 views

Show that matrices are not similar

I have to show that the following matrices are not similar: $$A = \left[\begin{matrix} 1 & 3 & -3 \\ -3 & 7 & -3 \\ -6 & 6 & -2\end{matrix}\right]$$ and $$A' = ...
0
votes
1answer
43 views

Hessian Matrix and critical point

Consider $f(x,y,z)\in C^2$. Suppose that $(0,0,0)$ is a critical point of $f$ and the Hessian Matrix of $f$ in $(0,0,0)$ is given by $\left(\begin{array}{ccc} 1 & 0 & \pi\\ 0 & \omega ...
1
vote
2answers
55 views

question in linear algebra on Hermitian matrices

Hello this indeed a very short question from Algebra that I have no real idea on and figured it is simple but for some reason I cannot seem to find it. I am given $A$ and $B$ complex square matrices ...
6
votes
2answers
9k views

Column Vectors orthogonal implies Row Vectors also orthogonal?

If the column vectors of a matrix $A$ are all orthogonal and $A$ is a square matrix, can I say that the row vectors of matrix $A$ are also orthogonal to each other? From the equation $Q \cdot ...
0
votes
0answers
28 views

Matrix of node's ranking in a graph

In any graph of n nodes in any dimension, define matrix $M_r$ of ranking as $\forall r_{ij}\in M_r$, $r_{ij}$ is the ranking of j to i. That is, j is the $r_{ij}$th nearest node to i. Therefore, any ...
0
votes
0answers
23 views

matrix input convergence

I'm hesitant to ask this here as I'm relatively new to matrix mathematics, so please go easy on me. I have an input vector of size n to a matrix of size ...
3
votes
1answer
56 views

Difficulties understanding these statements about change of basis

I understood more or less what a change of basis matrix is and how I can use it to pass to one coordinate system to another. Basically, a change of basis matrix is a matrix whose columns are the ...
0
votes
2answers
773 views

Find the value of $k$ for which matrix is diagonalizable

Consider the matrix $$A = \begin{bmatrix} 1 & 0 & 1 \\ 0 & 2 & k \\ 0 & 0 & 2 \\ \end{bmatrix}$$ where $k$ is a real number. The ...
0
votes
5answers
218 views

$AB-BA=A$, Then A is singular? [duplicate]

Title is the question, I tried taking trace both side and got trace of $A$ is zero, now to conclude $A$ is singular, suppose $A$ is non singular, then multiplying both side by Inverse of $A$ we get ...
1
vote
1answer
46 views

Determinants using elementary row operations

Let matrix $A$ be defined as \begin{pmatrix} a_{11} & a_{12} & \cdots & a_{1n} \\ a_{21} & a_{22} & \cdots & a_{2n} \\ \vdots & \vdots & \vdots & \vdots \\ ...
0
votes
0answers
58 views

Which classes of matrices contain $A$ and which contain $B$?

14. In the list below, which classes of matrices contain $A$ and which contain $B$? $$ A = \begin{bmatrix} 0 & 1 & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 ...
23
votes
3answers
1k views

Are $10\times 10$ matrices spanned by powers of a single matrix?

I don't know how to answer this question: Is there a $10 \times 10$ matrix $A$ such that $$M_{10}(\mathbb{F})=\text{span}\{I,A,A^2,\ldots, A^{100}\}\textrm{,}$$ where $M_{10}(\mathbb{F})$ is the ...
0
votes
2answers
32 views

Orthogonal Transformations and Eigenvalues

True or false: There exists and orthogonal matrix $T$ that has 2 as an eigenvalue. I think this is false, but I do not know how to prove it
0
votes
3answers
56 views

Diagonizable matrix

Got this matrix: \begin{bmatrix} 1 & 2 \\ -2 & 5 \end{bmatrix} I should determine if the matrix is diagonalizable or not. I found the eigenvalues ( only one) = 3. My eigenvector is then ...
0
votes
1answer
53 views

Magnitude of product of symmetric matrix and unit vector

If $A$ is any symmetric 2 by 2 matrix with eigenvalues -3 and 3 and $\vec{u}$ is a unit vector in $\mathbb{R}^2$, what is $||A\vec{u}||$? Any help would be appreciated, I haven't the slightest idea ...
2
votes
0answers
141 views

Norm of the inverse of a tridiagonal

Let's take a tridiagonal matrix (in general not Toeplitz, nor symmetric) $$L=\begin{pmatrix}a_1 & -b_1 & & & \\ -c_1 & a_2 & -b_2 \\ & -c_2 & \ddots & \ddots\\ ...
1
vote
1answer
39 views

What does it mean to take the power of a transition matrix? Or multiply it by a vector?

I understand the general idea that a matrix $M$ has some cell $M_{ij}$ that denotes the number of ways we can go from state $i$ to state $j$, but what does $(M^t)_{ij}$ represent? The number of ways ...
6
votes
2answers
193 views

Efficient way to compute $(A+D)^{-1}$ when $A^{-1}$ is known

I need to compute the inverse of a matrix sum $A+D$, where the inverse of $A\in\mathbb{R}^{n\times n}$ is known. The matrix $D\in\mathbb{R}^{n\times n}$ is a diagonal matrix which can be thought of as ...
2
votes
2answers
87 views

Inverse of a matrix with $a+1$ on the diagonal and $a$ in other places

Let $a>0$. Let $A$ be the $n\times n$ matrix with $a+1$ on the diagonal and $a$ in all other entries. How can one compute $A^{-1}$ as a function of $n$?
2
votes
1answer
92 views

Is there a way to directly solve this matrix equation: $XAX^{T} = B$

$X^{T}$ is the transpose of $X$. $A$ is a $n$ x $n$ matrix and $B$ is a $m$ x $m$ matrix, $m$ > $n$, both of them are known, $A$ is positive definitive and $B$ is symmetric. I would like to find $X$. ...
0
votes
1answer
45 views

Properties of Positive Semidefinite Matrices.

I am looking to do some modeling involving matrices and a requirement is the matrices be positive semidefinite and complex. However, the modeling tool I'm using does not handle complex values well, so ...
1
vote
3answers
73 views

Book on linear algebra containing interesting problems

Could anyone suggest me a problem book on linear algebra that contains interesting problems on rank, nullity, nullspace, linear transformations, eigenvalues, eigenvectors and characteristic ...